From: sbharris@ix.netcom.com(Steven B. Harris)
Newsgroups: sci.chem,sci.physics,sci.misc,sci.bio,sci.cryonics
Subject: Re: Freezing multicellular organisms
Date: 2 Jul 2000 06:24:01 GMT

In <8jltbe$e85$1@slb7.atl.mindspring.net> sbharris@ix.netcom.com(Steven
B. Harris) writes:
>
>In <395E51DB.CA31D492@eagle.ca> Graham Cowan <gcowan@eagle.ca> writes:

>>
>>Edward Green wrote:
>>
>>> >Since liquid helium II superleaks through channels
>>> >no liquid of nonzero viscosity can,
>>> >you may not need to do anything more than
>>> >leap into a big enough tub of it.
>>> >It will quickly perfuse you,
>>> >and your heat will quickly disperse throughout the tub,
>>> >and all your aqueous media will be far below their
>>> >glass transition temperature before you can say Jack.
>>>
>>> Are you certain?  I thought one of the features of liquid helium was a
>>> ridiculously low heat of boiling... there is going to be a whole lot
>>> of boiling going on when you jump into the pool, and this may
>>> disappointingly lower your freezing rate.
>>
>>I was thinking of rather a large tub,
>>such that all the heat that would exit a 100kg piece of water
>>as it cooled from 310K to 1.5K
>>would only raise the bath temperature from 0.5K to 1.5K.
>
>  Okay, sport, and how much would that be? My own back of envelope
>calculation follows:
>
>  It costs roughly 200 kcal/kg to cool water from body temp to absolute
>zero (including heat of fusion, and correction for loss of heat
>capacity in ice at low temperatures-- it takes about as much heat to
>melt ice as it gives up going from 0 C to 0 K). For a 100 kg person who
>is half water, call it 10,000 kcal total.  You don't want to warm your
>superfluid helium only to 1.5 K, since most of its heat capacity lies
>between 1.8 and 2.2 C.  I estimate the whole integrated heat capacity
>from O K to transition at 2.2 K at maybe 1.6 kcal/kg. No more than 2
>kcal/kg, certainly.  That means you need 5,000 kg superfluid helium, at
>density 147 kg/m^3. That's 34 m^3 of superfluid He, or a pool at least
>10 ft on a side and 10 ft deep.  Interesting-- I had in mind offhand
>something more swimming pool-sized.  As to whether or not you'd get
>boiling, that also is interesting.  The conductivity of liquid He is
>high but not infinite. The answer depends on how high it is.




  Some further comments about jumping into superfluid liquid
helium. The stuff is good at getting into things, but not magic.
I doubt very much if it can penetrate frozen tissues.  If you
could flush the circulatory system with helium gas and then keep
it open to the outside, you might get some improvement from
dunking in superfluid helium (it would be great at slipping
around bubble emboli).  Also, if you spiked the tissues with
metal conduction strips like aluminum nails in a baked potato
(another nano-robot repair job afterward-- no problem), you might
get really good results.  Otherwise, if you jump into superfluid
helium, you're going to cool by surface conduction only.

  Even if a man who jumps into a 10 ft^3 pool of superfluid
helium at 0 K attains and keeps a surface temperature of less
than 2 K (assuming perfect helium heat conduction and no
boiling), the man's cooling rate and core temp will be set by the
insulating quality of his tissues. A "rind" of frozen tissue and
another rind of unfrozen tissue stands between the heat in his
core, and his skin (which will be at bath temperature).  As we
shall see, it's mostly the un-solidified tissues that provide the
insulating power.

  The max at which a man can be cooled with superfluid He at a
gradient of 310 K is unknown, but some estimates may be made.  An
upper bound for efficiency can be calculated by simply assuming
that the surface of the body can be kept at the helium superfluid
temp, and then using results from a standard conductive heat
transfer equation, such as the Laplace equation, for a solid:

dT/dt = -D del T

Solving the Laplace equation conveniently for a "cubical" torso
chunk with mass 27,000 kg and linear dimension L = 30 cm on a
side, the temperature T(t) in the center of the body for t >>
L^2/D is given by:

T(t)= T(bath) * [1 - (4/pi)^3 exp-(t*3pi^2*D/L^2) ]

Using the thermal diffusivity D of biological tissues (about 1 x
10^-3 cm^2/sec), the exponential time constant -k = -3pi^2*D/L^2
= 3e-5 corresponds to a half-time (= ln2/k) of about 2000
seconds, but because of the (4/pi)^3 = 2.06 factor, the effective
half-time is only about ln2*(2000)= 1400 seconds = 22 min.  Thus,
going from an initial temperature differential of 310 K to half
this differential, or 155 K (about -118 C, a reasonable glass
vitrification storage temp) would require one half-time, or 22
minutes. This implies that it takes this long to reach vitrifi-
cation temps in a the body core.

  Assuming no boost from the better conduction of already
solidified tissues, that is-- but that's the rub.  The above uses
the thermal diffusivity of unfrozen biological tissues at
biological temperatures, which is not quite fair because after a
while, some of the tissue vitrifies, and after that, conduction
across it is faster, so the core of the man in superfluid helium
is insulated less. And the effect is larger, even, if there is
freezing.

  To see how much faster things go with freezing, if one starts
with the thermal diffusivity of ice (about 10 times that of
unfrozen tissue, due to the superior conductivity of solids) you
get a half-time of 2.2 minutes instead of 22 in the above
equation. But this isn't fair either, of course, because the 2
minute time assumes everything is frozen to start with. Also, I
would guess that frozen tissue probably has only half the
diffusivity of pure ice, so the half-life for cooling a solid
already frozen meat cube of 30 cm radius by perfect surface
conduction in liquid helium, is probably closer to 4 minutes.

  The problem, of course, lies in the solution of the Laplace
equation used above, which assumed that the diffusivity of the
medium did not change, and that there was no heat of fusion
source-term.  Both of these are gross errors, for even if there
is no heat of fusion (as in tissue vitrification), the
conductivity of the tissue will increase at least several-fold as
it vitrifies (to a value midway somewhere between ice and water),
and also the tissue heat capacity for any system (frozen OR
vitrified) will fall drastically (due to quantum effects) as it
approaches near-zero temperatures.  Both of these effects act to
increase thermal diffusivity D at cryogenic temperatures over
that of normal tissues (D is the ratio of conductivity and volume
specific heat capacity). Thus, we greatly overestimate the time
tissues take to reach liquid helium temps if we use normal tissue
D, but it's hard to say what to do instead.  It is for this
reason that most heat conduction problems require numerical
modeling, and can't be done analytically.

  What the hell--here's a guess: If a man has not been prepared
with cryoprotectants, so that he's going to freeze and not
vitrify when he goes into superfluid helium, we can make a gross
estimate that, starting from solid ice at 273 K, it takes less
than 6 half-times of 4 minutes to get his core to liquid helium
temperatures. This is 24 minutes. We might guess that the real
value is probably less by several half-times, because of the
lower heat capacity for the cryogenic temperatures characteristic
of the last few half-times. Call it 12-16 min at a guess.

  So how long to solid ice? As he freezes at first, his tissues
will be composed of an expanding ice shell which will have a
constant 273 K temperature gradient across it, because the
outside is at O Kelvins and the shrinking inside can't be lower
than 273 Kelvins, since it's still liquid. (I will assume that
the time for all the water in the core to cool to 273 is fairly
short, and that this cooling doesn't make much difference
anyway.)  If we figure a constant 273 K gradient across an ice-
insulated sphere, with the ice insulation growing thicker by an
amount equal to ice mass being made, which is in turn the total
heat conducted across the ice divided by the heat of fusion of
water, I figure that it will take about 3.5 minutes to freeze the
center of a meat sphere 30 cm in radius, in superfluid helium.
Unless I've slipped a factor of pi or 2.

  Putting these figures together I get an estimate of the answer:
It takes 3.5 minutes for a man in superfluid helium at 0 K to
freeze to the core, and less than 12-16 min after that to take
the ice down to below liquid helium temps.  Of course, at the
moment the last water freezes, a good deal of cooling of the
outer ice has already been done, so again the time for this is
less that we would estimate from starting at ice at 273 C. My
final guess is that the whole thing would take about 15 min. How
long you scream for, I don't know.  Less.

  For vitrification, it's more complicated, not least because I
don't know the thermal diffusivity of vitrified tissue. Assuming
that it's 3 times that of normal tissue gets us a half-time of
between 22 and 7 minutes, to get the core halfway to the initial
310 K temp difference, or down to 155 K = -118 C, which will
vitrify the core. At a guess, about 15 minutes to vitrify.
Interestingly, it seems under these assumptions to take the SAME
to vitrify at half the way to the temp of liquid helium, as it
does to *freeze* ALL the way to the temp of liquid helium. The
boost from ice conduction is the difference.